1. Field of the Invention
The invention relates to the field of electrical power systems, and more particularly to methods for on-line transient stability analysis, on-line dynamic security assessments and energy margin calculations of practical power systems.
2. Description of the Related Art
By nature, power systems are continually experiencing disturbances which may cause power system instability. These disturbances can be classified as either event disturbances or demand disturbances. Event disturbance can be a short-circuit fault, or the loss of a generator, load or transmission line facility, or a combination of the above. Demand disturbance can be load variations at a set of buses, or power transfer between two sets of specified buses, or a combination of these two. Several recent power system blackouts due to disturbances have occurred in several countries, including Belgium, Canada, France, Japan, Sweden and the United States. Modern energy management systems typically do not perform on-line dynamic security assessment (DSA) to ensure the ability of the power system to withstand certain credible contingencies (disturbances). As our society is increasingly dependent on reliable electricity supply and blackouts are becoming more costly when they occur, any violation of stability limits can have huge impacts (financially and physically) on society. Especially in the era of de-regulation, on-line DSA is an important tool that is needed to avoid any potential blackout.
Power systems must be planned and operated to withstand the occurrence of certain credible disturbances. At present, modern energy management systems (EMS) only perform the task of on-line static security assessment but not the task of on-line dynamic security assessment. Hence, modern EMS still can not assess the ability of a power system to withstand credible contingencies (disturbances.) The set of credible contingencies is a collection of disturbances that are likely to occur with potentially serious consequences. The extension of EMS to include on-line dynamic security assessment (DSA) is desirable and is becoming a necessity for modern power systems. This extension is, however, a rather challenging task; despite the consistent pressure for such an extension, partly due to economic incentives and partly due to environmental concerns, performing DSA has long remained an off-line activity. Indeed, from a computational viewpoint, on-line static security assessment needs to solve a large set of nonlinear algebraic equations. On-line dynamic security assessment (DSA) however requires the handling of a large set of nonlinear differential equations in addition to the nonlinear algebraic equations involved in the SSA. The computational effort required in on-line DSA is roughly three magnitudes higher than that for the SSA.
At present, transient stability analysis programs routinely used in utilities around the world are based exclusively on step-by-step numerical techniques. This practice of power system transient stability via the time-domain approach has a long history. Although the time-domain approach, by its nature, has several advantages, it has several disadvantages. For example, the time-domain approach requires intensive, time-consuming computation efforts; therefore the time-domain approach has not been suitable for on-line application. The time-domain approach does not provide information regarding the degree of stability/instability, or how far the system is from transient instability. This piece of information is valuable for both power system planning and operations. Furthermore, the time-domain approach does not provide information as to how to derive enhancement control or preventive control actions for maintaining system stability.
On-line DSA offers multiple engineering and financial benefits. Some examples are listed below: (i) it can avoid potential blackouts, (ii) a power system can be operated with operating margins reduced by a factor of 10 or more if the dynamic security assessment is based on actual system configurations and actual operating conditions, instead of assumed worst-case conditions, as is done in off-line studies. On-line DSA provides such a capability, which is especially significant in that the demands on current power system environments push the operating conditions of power systems not only with low reserve margins but also closer to their stability limits. (iii) On-line DSA can lead to financial benefits. For instance, it can provide accurate transfer capability constrained by transient stability limit. This accurate calculation of transfer capability would allow remote generators with low production cost to be economically dispatched. The cost saving can be significant, e.g. $300K/day, for a mid-size power system.
From a functional requirements viewpoint, on-line DSA must provide the following                Fast stability assessments of a list of credible contingencies        Accurate identification of unstable contingencies which has no post-fault steady-state (Contingencies Lead to System Collapse)        Accurate identification of unstable contingencies which has negative energy margins (Contingencies lead to system transient instability)        Accurate identification of severe contingencies (with small but positive energy margins)        Contingency screening and ranking for transient stability in terms of energy margin or critical clearing time        Detailed time-domain simulations of selected unstable or severe contingencies (After the initial event of the contingency, the system variables such as rotor angles, rotor frequencies, voltages, currents, power flows should be simulated as the system responds to controls and protection schemes and to other possible operational events such as arm/disarm special protection schemes, enable/disable control functions, or supervisory switching actions)        
To significantly reduce the computational burden required for on-line DSA, the strategy of using an effective scheme to screen out a large number of stable contingencies and to only apply detailed simulation programs to potentially unstable contingencies is well recognized. This strategy has been successfully implemented in on-line SSA and can potentially be applied to on-line DSA. Given a set of credible contingencies, the strategy would break the task of on-line DSA into two assessment stages:                Stage 1: Perform the task of fast dynamic contingency screening to screen out contingencies which are definitely stable from a set of credible contingencies        Stage 2: Perform a detailed stability assessment and energy margin calculation of each contingency remaining after Stage 1.        
Several research developments in on-line dynamic contingency screening have been reported in the literature. At present, the existing methods for dynamic contingency screening all first perform extensive numerical simulation on a set of credible contingencies using off-line network data in order to capture essential stability features of system dynamical behaviors; they then construct a classifier attempting to correctly classify contingencies on new and unseen network data in an on-line mode. These methods cannot meet the on-line computation requirement as well as reliability requirement.
Recently, a systematic method to find the controlling unstable equilibrium point was developed, called the BCU method, and is disclosed in U.S. Pat. No. 5,483,462 to Chiang [1]. However, it has been found in several numerical studies that the BCU method may fail in the sense that the unstable equilibrium point (UEP) computed by the BCU method may not always lie on the stability boundary of the original post-fault system. Thus, the energy value at the computed UEP which does not lie on the stability boundary of the original post-fault system can not be used as a critical energy for direct stability assessment. Recently, a set of BCU classifiers for the on-line dynamic contingency screening of electric power systems was developed and disclosed in U.S. Pat. No. 5,719,787 to Chiang and Wang [2]. However, numerical simulation results indicate that the BCU classifiers may mis-classify unstable contingencies as stable. For instance, 10 unstable contingencies in a 173-bus power system were mis-classified as stable; hence violating the reliability requirement of a dynamic security classifier.
A set of several improved BCU classifiers for the on-line dynamical security screening of practical power systems were developed, and is disclosed in [3]. The improved BCU classifiers not only meet the five requirements described in [2] for on-line dynamical security assessments. Furthermore, improved BCU classifiers computes energy margins for screened stable contingencies.
At the present time, the only factor that degrades the reliability of the BCU method is that the controlling UEP computed by the BCU method may not always be the true (correct) controlling UEP. However, since the one-parameter transversality conditions, which lie at the heart of the BCU method, are not easily verifiable, one cannot guarantee a 100-percent reliability for the correctness of the CUEPs computed by the BCU method. Hence, new techniques are required which can not only bypass the difficulty of verifying the one-parameter transversality conditions but can also improve the reliability of the BCU method. In this invention, we will present the development of our invented method, termed group-based BCU method, which is to enhance both the reliability and accuracy of the BCU method in computing critical energy values.
We will begin our discussion by introducing the so-called one-parameter transversality conditions on which the theoretical basis of the BCU method is built. Following this, an important issue regarding the verification of the correctness of CUP's computed by the BCU method will be addressed analytically.
2.1 One-Parameter Transversality Condition
In the BCU method, the one-parameter transversality condition is a sufficient condition to ensure that the UEP computed by the BCU method lies on the stability boundary of the (post-fault) power system. We point out that the one-parameter transversality condition is not a necessary condition and that the development of a numerical procedure to check the one-parameter transversality condition can be very involved and may be unnecessary. We propose to take another approach to verify whether the UEP computed by the BCU method lies on the stability boundary of the (post-fault) power system. To explain this approach, we first review the one-parameter transversality condition in the BCU method.
In developing a BCU method for a given power system stability model, the associated artificial, reduced-state model must be defined. To explain the reduced-state model, we consider the following generic network-preserving transient stability model,
                              0          =                                                                      ∂                  U                                                  ∂                  u                                            ⁢                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                      +                                          g                1                            ⁡                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                                    ⁢                                  ⁢                  0          =                                                    -                                                      ∂                    U                                                        ∂                    w                                                              ⁢                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                      +                                          g                2                            ⁡                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                                    ⁢                                  ⁢                              T            ⁢                                                  ⁢                          x              .                                =                                                    -                                                      ∂                    U                                                        ∂                    x                                                              ⁢                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                      +                                          g                3                            ⁡                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                                    ⁢                                  ⁢                              y            .                    =          z                ⁢                                  ⁢                              M            ⁢                                                  ⁢                          z              .                                =                                    -              Dz                        -                                                            ∂                  U                                                  ∂                  y                                            ⁢                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                      +                                          g                4                            ⁡                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                                                        (        1        )            where U(u,w,x,y) is a scalar function. Regarding the original model (1), we choose the following differential-algebraic system as the artificial, reduced-state model.
                              0          =                                                    -                                                      ∂                    U                                                        ∂                    u                                                              ⁢                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                      +                                          g                1                            ⁡                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                                    ⁢                                  ⁢                  0          =                                                    -                                                      ∂                    U                                                        ∂                    w                                                              ⁢                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                      +                                          g                2                            ⁡                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                                    ⁢                                  ⁢                              T            ⁢                                                  ⁢                          x              .                                =                                                    -                                                      ∂                    U                                                        ∂                    x                                                              ⁢                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                      +                                          g                3                            ⁡                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                                    ⁢                                  ⁢                              y            .                    =                                                    -                                                      ∂                    U                                                        ∂                    y                                                              ⁢                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                      +                                          g                4                            ⁡                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                                                        (        2        )            
The fundamental ideas behind the BCU method can be explained as follows. Given a power system stability model (which admits an energy function), the BCU method first explores the special properties of the underlying model with the aim of defining an artificial, reduced-state model such that the following static as well as dynamic relationships are met.
Static Properties
                (S1) the locations of equilibrium points of the reduced-state model correspond to the locations of equilibrium points of the original model (1). For example, (û,ŵ,{circumflex over (x)},ŷ) is an equilibrium point of the reduced-state model if and only if (û,ŵ,{circumflex over (x)},ŷ,0) is an equilibrium point of the original model (1), where 0εRm and m is an appropriate positive integer,        (S2) the types of equilibrium points of the reduced-state model are the same as that of the original model. For example, (us,ws,xs,ys) is a stable equilibrium point of the reduced-state model if and only if (us,ws,xs,ys,0) is a stable equilibrium point of the original model. (û,ŵ,{circumflex over (x)},ŷ) is a type-k equilibrium point of the reduced-state model if and only if (û,ŵ,{circumflex over (x)},ŷ,{circumflex over (0)}) is a type-k equilibrium point of the original model.Dynamical Properties        (D1) there exists an energy function for the artificial, reduced-state model (2).        (D2) an equilibrium point, say, (û,ŵ,{circumflex over (x)},ŷ) is on the stability boundary ∂A(us,ws,xs,ys) of the reduced-state model (2) if and only if the equilibrium point (û,ŵ,{circumflex over (x)},ŷ,0) is on the stability boundary ∂A(us,ws,xs,ys,0) of the original model (1)        (D3) it is computationally feasible to efficiently detect the point at which the projected fault-on trajectory (u(t),w(t),x(t),y(t)) hit the stability boundary ∂A(us,ws,xs,ys) of the post-fault reduced-state model (2) without resorting to an iterative time-domain procedure to compute the exit point of the post-fault reduced-state model (2).        
The dynamic relationship (D3) plays an important role in the development of the BCU method to circumvent the difficulty of applying an iterative time-domain procedure to compute the exit point on the original model. The BCU method then finds the controlling UEP of the artificial, reduced-state model (2) by exploring the special structure of the stability boundary and the energy function of the reduced-state model (2). Next, it relates the controlling UEP of the reduced-state model (2) to the controlling UEP of the original model (1).
Given a power system stability model, there exists a corresponding version of the BCU method. The BCU method does not directly compute the CUEP of the original model because computing the exit point of the original model, which is a key to computing the controlling UEP, requires an iterative time-domain procedure. Instead, the BCU method computes the CUEP of the original model (1) via computing the CUEP of the artificial, reduced-state model (2).
We next present some analytical results showing that, under certain conditions, the original model (1) and the artificial, reduced-state model (2) satisfy static relationships (S1) and (S2) as well as dynamic relationships (D1) and (D2). A computational scheme will be developed and incorporated into the BCU method to satisfy dynamic relationship (D3).
Theorem 1: (Static Relationship)
Let (us,ws,xs,ys) be a stable equilibrium point of the reduced-state model (2). If the following conditions are satisfied:                (1) zero is a regular value of        
            ∂      4        ⁢          U      ⁡              (                              u            i                    ,                      w            i                    ,                      x            i                    ,                      y            i                          )                        ∂      u        ⁢          ∂      w        ⁢          ∂      x        ⁢          ∂      y      for all the UEP (ui,wi,xi,yi) i=1, 2, . . . , k on the stability boundary ∂A(us,ws,xs,ys),                (2) the transfer conductance of reduced-state model (2) is sufficiently small,Then, (û,ŵ,{circumflex over (x)},ŷ) is a type-k equilibrium point of reduced-state model (2) if and only if (û,ŵ,{circumflex over (x)},ŷ,0) is a type-k equilibrium point of the original model (1).        
Theorem 1 asserts that, under the stated conditions, the static properties (S1) and (S2) between original model (1) and the reduced-state model (2) hold.
It can be shown that there exists a numerical energy function for the reduced-state model (2). More specifically, it can be shown that for any compact set S of the state-space of model (2), there is a positive number α such that, if the transfer conductance of the model satisfies |G|<α, then there is an energy function defined on this compact set S.
To examine the dynamic property (D2), we introduce the following family of one-parameterized systems d(λ).
                                                        ɛ              1                        ⁢                          u              .                                =                                    -                                                ∂                  U                                                  ∂                  u                                                      ⁢                          (                              u                ,                w                ,                x                ,                y                            )                                      ⁢                                  ⁢                                            ɛ              2                        ⁢                          w              .                                =                                    -                                                ∂                  U                                                  ∂                  w                                                      ⁢                          (                              u                ,                w                ,                x                ,                y                            )                                      ⁢                                  ⁢                              T            ⁢                                                  ⁢                          x              .                                =                                    -                                                ∂                  U                                                  ∂                  x                                                      ⁢                          (                              u                ,                w                ,                x                ,                y                            )                                      ⁢                                  ⁢                              y            .                    =                                                                      (                                      1                    -                    λ                                    )                                ⁢                z                            -                              λ                ⁢                                                                  ⁢                                  y                  .                                                      =                                          -                                                      ∂                    U                                                        ∂                    y                                                              ⁢                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                                    ⁢                                  ⁢                              M            ⁢                                                  ⁢                          z              .                                =                                                    -                Dz                            -                                                (                                      1                    -                    λ                                    )                                ⁢                z                            -                              λ                ⁢                                                                  ⁢                                  y                  .                                                      =                                          -                                                      ∂                    U                                                        ∂                    y                                                              ⁢                              (                                  u                  ,                  w                  ,                  x                  ,                  y                                )                                                                        (        3        )            Theorem 1: (Dynamic Relationship)
Let (us,ws,xs,ys) be a stable equilibrium point of the reduced-state model (2). If the following conditions are satisfied,                (1) zero is a regular value of for all the UEP on the stability boundary.        (2) the transfer conductance of the reduced-state model (2) is sufficiently small,        (3) all the intersections of the stable and unstable manifolds of the equilibrium points on the stability boundary ∂A(us,ws,xs,ys,0) of the one-parameterized model d(λ) (3) satisfy the transversality condition for λε[0,1],then:        [1] the equilibrium point (ui,wi,xi,yi) is on the stability boundary ∂A(us,ws,xs,ys) of model (2) if and only if the equilibrium point (ui,wi,xi,yi,0) is on the stability boundary ∂A(us,ws,xs,ys,0) of model (1)        [2] the stability boundary ∂A(us,ws,xs,ys) of model (2) is the union of the stable manifold of all the equilibrium points (ui,wi,xi,yi), i=1, 2, . . . , on the stability boundary ∂A(us,ws,xs,ys); i.e.∂A(us,ws,xs,ys)=∪Ws(ui,wi,xi,yi)  (4)        [3] the stability boundary ∂A(us,ws,xs,ys,0) of model (1) is the union of the stable manifold of all the equilibrium points (ui,wi,xi,yi,0), i=1, 2, . . . , on the stability boundary ∂A(us,ws,xs,ys,0); i.e.∂A(us,ws,xs,ys,0)=∪Ws(ui,wi,xi,yi,0)  (5)        
Theorem 1 asserts that, under the stated conditions, conditions (1)-(3), the dynamic property (D2) is satisfied. Furthermore, the stability boundaries of both models are completely characterized. Condition (1) is a generic property while conditions (2) and (3) are not. We will present an approach to check the dynamic property (D2) without checking conditions (2) and (3).
A Conceptual Network-Preserving BCU Method
Theorem 1 and Theorem 1 provide the theoretical basis for finding the controlling UEP of the original is network-preserving model (1) via the controlling UEP of the artificial, reduced-state model (2). A conceptual BCU method for the network-preserving model is presented in the following:
A Conceptual BCU Method for the Network-Preserving Model                Step 1. From the (sustained) fault-on trajectory (u(t),w(t),x(t),y(t),z(t)) of the network-preserving model (1), detect the exit point (u*,w*,x*,y*) at which the projected trajectory (u(t),w(t),x(t),y(t)) of the network-preserving model exits the stability boundary of the post-fault reduced-state model (2).        Step 2. Use the point (u*,w*,x*,y*) as the initial condition and integrate the post-fault reduced-state model (2) to find the UEP whose stable manifold contains the exit point (u*,w*,x*,y*).        Step 3. The controlling UEP with respect to the fault-on trajectory of the network-preserving model (1) is (uco*,wco*,xco*,yco*,0)        
The essence of the BCU method is to compute the controlling UEP of the original model (1) via computing the controlling UEP of the reduced-state model (2) whose controlling UEP can be effectively computed. Step 1 and Step 2 of the conceptual BCU method find the controlling UEP of the reduced-state model (i.e. the controlling UEP of the projected fault-on trajectory). Step 3 relates the controlling UEP of the reduced-state model (2) to the controlling UEP of the original model (1). BCU method does not perform its calculation of CUEP in the state-space of the underlying original power system transient stability model. This is because the task of computing the exit point of the original model, which is a key to computing CUEP, requires an iterative time-domain procedure. Instead, BCU method computes the CUEP of the original model via computing the CUEP of an artificial reduced-state model. As such, BCU method computes CUEP with varying degree of success. The UEP computed by the BCU method may not always be the CUEP.
The one-parameter transversality conditions play an important role in the theoretical foundation of the conceptual BCU method. The violation of the one-parameter transversality conditions may cause incorrectness in the BCU method when computing the controlling UEP. However, due to the complexity of practical power system models, the one-parameter transversality conditions may not be always satisfied. There are several counter-examples which show the BCU method may fail to give correct stability assessments.
Based on the above analysis, we will take a different approach. Instead of checking the one-parameter transversality condition and the small-transfer-conductance condition, we propose to directly check whether or not the UEP (uco*,wco*,xco*,yco*,0) lies on the stability boundary of the original model; i.e. check the dynamic property (D2) directly. We will also term the dynamic property (D2) the boundary property.
It can be shown that the boundary property holds for high damping systems while it may not hold for low damping systems. The issue of how to determine the critical damping value above which the boundary property holds remains open. The critical damping value seems to depend on a variety of factors including network topology, loading condition, and system models used, among others.
Damping Terms and Boundary Property
It has been shown that under the one-parameter transversality condition, BCU method can compute exact CUEP. However, the verification of one-parameter transversality condition is not an easy task either. At the present time, with the development of improved BCU classifiers, the only factor that degrades the reliability of the BCU method (i.e. BCU method gives incorrect stability assessments) is that the unstable equilibrium point (UEP) computed by the BCU method may not always be the true (correct) controlling UEP. Furthermore, UEPs computed by BCU method may not even satisfy the boundary condition. We say a computed UEP (with respect to a contingency) is said to satisfy the boundary condition if the computed UEP lies on the stability boundary of the original post-contingency system.
This factor can clearly explain the reason why the BCU method may give incorrect stability assessments for certain types of contingencies. BCU method fails because of the violation of the boundary condition due to insufficient system damping. Technically speaking, insufficient system damping leads to the occurrence of global bifurcation in the parameterized dynamical systems underlying the BCU method. On the other hand, it has been found that the BCU method performs very well if the boundary condition is satisfied; in addition, the boundary condition is satisfied if the system damping terms are sufficiently large.
The BCU method may give incorrect stability assessments for certain types of contingencies (cases). For illustrative purpose, we will present numerical results in which the BCU method fails to give correct stability assessments for some cases, due to either multi-swing phenomenon or light damping. In all cases, the BCU method fails because of the violation of boundary property. It will be shown that the BCU method works well if the boundary property is satisfied; furthermore, the boundary property is satisfied if the system damping terms are sufficiently large.
Some Numerical Examples
We apply the BCU method to analyze a contingency list of a test system. We present some cases in which the BCU method fails and point out the reasons why this failure occurs.
Table 1 displays some cases in which the BCU method fails to give correct stability assessments. We point out that these cases all exhibit multi-swing phenomena and the boundary properties are not satisfied. All the cases listed in Table 1 belong the same group; group #4. We also present some cases in which the BCU method works well in Table 2. It should be pointed out that all the cases in Table 2 belong to two groups of coherent contingencies, group #13 and group #43, and they all satisfy the boundary condition.
TABLE 1A group of coherent contingencies which the BCU Method Fails.FaultTime-BoundaryFaultClearingBCUBCUDomainMode ofDistance/BoundaryDescriptionTime(s)MarginAssessmentAssessmentInstabilityUEP GroupPropertyFault Bus 5360.071.1127StableStableMulti-swing0.234/4NoOpen line0.100.983StableUnstable0.234/4No536-537Fault Bus 7070.071.17StableStableMulti-swing0.234/4NoOpen line0.100.9433StableUnstable0.234/4No707-708Fault Bus 5210.050.3587StableUnstableMulti-swing0.232/4NoOpen line0.070.2387StableUnstable0.232/4No521-522Fault Bus 520.051.0874StableUnstableMulti-swing0.200/4NoOpen line0.070.8627StableUnstable0.200/4No52-575
TABLE 2A group of coherent contingencies which the BCU Method Succeeds.FaultTime-BoundaryClearingBCUBCUDomainMode ofDistance/BoundaryFault DescriptionTime(s)MarginAssessmentAssessmentInstabilityUEP GroupPropertyFault Bus 7090.070.948StableStableSingle-swing1.0/43YesOpen line 104-7090.10−0.225UnstableUnstable1.0/43YesFault Bus 380.070.468StableStableMulti-swing1.0/13YesOpen line 38-550.10−0.551UnstableUnstable1.0/13Yes
TABLE 3A group of coherent contingencies which the BCU Method Fails.FaultTime-BoundaryFaultClearingBCUBCUDomainMode ofDistance/BoundaryDescriptionTime(s)MarginAssessmentAssessmentInstabilityUEP GroupPropertyFault Bus 30360.050.0211StableUnstableMulti-swing0.200/27NoOpen line0.07−0.0695UnstableUnstable0.200/27No3036-3037Fault Bus 40210.051.9706StableUnstableMulti-swing0.310/39NoOpen line0.071.9420StableUnstable0.310/39No4021-4022Fault Bus 40290.070.5595StableUnstableMulti-swing0.289/35NoOpen line0.27−0.3527UnstableUnstable0.289/35No4026-4029Fault Bus 1070.070.4302StableUnstableMulti-swing0.234/52NoOpen line 107-41240.15−0.1049UnstableUnstable0.234/52NoThe Damping Factors
We have observed that the reliability of BCU method in stability assessments increases (i.e., it works on more number of contingencies) as the system damping factors become larger. For example, while BCU method fails on contingencies as listed in Table 3 due to small damping, the method works on some of these contingencies, in particular on the group #52 if the system damping factor increases, as shown in Table 4. It can be seen from this table that the boundary distance becomes closer to 1.0; in other words, the computed CUEP becomes closer to the stability boundary of the original system. Furthermore, the time-domain behaviors of the study power system subject to these contingencies improve; more specifically, the transient stability is enhanced.
As we further increase the damping effect, the reliability of the BCU method is further improved. Table 5 lists the performance of the BCU method for the same cases as those in Table 4, except that the system damping factors are doubled. BCU method computes CUEPs satisfying the boundary property for these contingencies in groups #27, #39 and #52 of coherent contingencies. In addition, the boundary distance of the computed UEP relative to each contingency in group #35 all lies closer to 1.0. These results clearly show that as the system damping factor increases, the boundary distance of the computed CUEP by the BCU method increases and the satisfiability of boundary condition relative to contingencies by the BCU method also increases. Furthermore, the transient stability of the study power system subject to these contingencies is enhanced.
TABLE 4BCU method works for group #52 when damping is increased.FaultTime-BoundaryClearingBCUBCUDomainMode ofDistance/BoundaryFault DescriptionTime(s)MarginAssessmentAssessmentInstabilityUEP GroupPropertyFault Bus 30360.070.177StableStableMulti-swing0.701/27NoOpen line0.17−0.303UnstableUnstable0.701/27No3036-3037Fault Bus 40210.072.727StableStableMulti-swing0.824/39NoOpen line0.77−0.157UnstableUnstable0.824/39No4021-4022Fault Bus 40290.070.563StableStableMulti-swing0.867/35NoOpen line0.37−0.048UnstableUnstable0.867/35No4026-4029Fault Bus 1070.071.081StableStableMulti-swing 1.00/52YesOpen line0.30−0.245UnstableUnstable 1.00/52Yes107-4124
TABLE 5BCU method works for groups #27, #39 and #52 when damping is increased.FaultTime-BoundaryClearingBCUBCUDomainMode ofDistance/BoundaryFault DescriptionTime(s)MarginAssessmentAssessmentInstabilityUEP GroupPropertyFault Bus 30360.070.934StableStableMulti-swing1.00/27YesOpen line0.170.525StableStable1.00/27Yes3036-3037Fault Bus 40210.072.813StableStableMulti-swing1.00/39YesOpen line0.771.750StableStable1.00/39Yes4021-4022Fault Bus 40290.070.588StableStableMulti-swing0.948/35 NoOpen line0.370.279StableStable0.948/35 No4026-4029Fault Bus 1070.071.182StableStableMulti-swing1.00/52YesOpen line0.300.306StableStable1.00/52Yes107-4124
From the viewpoint of state space, the increase of the boundary distance of the computed UEP due to the increase of the system damping clearly demonstrates the effect of damping terms on transient stability; it enhances the transient stability by enlarging the stability region of the post-fault SEP, hence it increases the critical clearing times as well as the energy margin.
These observations will lead to the development of a group-based BCU method in which the boundary property will be checked. In order to develop schemes for an efficient check of the boundary property, the concept of a group of coherent contingencies will be proposed and explored. Several group properties will be explored and investigated. These group properties will be taken into the development of the group-based BCU method. The group-based BCU method will also include a scheme to compute the critical energy for those contingencies in which the boundary property is not satisfied.
With the introduction of a boundary property, one can check the correctness of a computed CUEP, say by the BCU method, through checking its boundary property; instead of checking the one-parameter transversality condition which is very difficult to check. By computing the boundary distance of the computed CUEP, one can verify whether or not the computed CUEP lies on the stability boundary of the original system; if the boundary of the computed CUEP is 1.0, then the CUEP lies on the stability boundary of the original post-fault system; otherwise, it is not.
It will be shown that the boundary property is a group property (a group property is a property which holds for every member in the group); hence it is not necessary to compute the boundary distance for each computed UEP in each group of coherent contingencies. Computing the boundary distance of a UEP in a group of coherent contingencies is sufficient to determine the boundary property of all of the contingencies in the group. The exploration of the group property will lead to a significant reduction in computation, as will be explained later.
We will describe in this invention a novel system, called Group-based BCU-DSA, for on-line dynamic security assessments and energy margin calculations of practical power systems in modern energy management systems. The novel system meets the requirements of on-line dynamic security assessment and energy margin calculations through effective exploration of the merits of both the group-based BCU method (and the improved BCU classifiers) and the detailed time-domain simulation program. There are three major components in this architecture: (i) a sequence of improved BCU classifiers whose major functions are to screen out, from a set of credible contingencies, all of those contingencies which are definitely stable and to capture all of the (potentially) unstable contingencies, (ii) a BCU-guided time-domain program for stability analysis and energy margin calculation of both the (potentially) unstable and undecided contingencies captured by the sequence of improved BCU classifiers in (i), and (iii) a group-based BCU method.